Timeline for what is the associated Borel set of a Borel measurable function on the extended real line?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| May 8, 2010 at 11:19 | vote | accept | zzzhhh | ||
| May 8, 2010 at 10:35 | comment | added | zzzhhh | Thank you very much Kevin! I found these references and managed to prove the equivalence of the top-down definition in Folland and Bottom-up definition in Bartle. I post the proof below for my future reference. Today is really a fruitful day, thank you all! PS: Should I continue Halmos or turn to Bartle for a better effect of self-study? | |
| May 8, 2010 at 6:49 | comment | added | Kevin Ventullo | In general, the Borel sets of any topological space are just the sigma-algebra generated by the opens. See, for example, pg. 22 of Folland's Real Analysis. Another good reference is "The Elements of Integration and Lebesgue Measure" by Bartle. On page 7 he describes precisely the example of the extended real line. | |
| May 7, 2010 at 9:47 | comment | added | zzzhhh | Thank you. This is a sigma-ring that can constitute a measurable space together with $\mathbb B^\ast$, and consideration of only $f^{-1}({+\infty})$ and $f^{-1}({-\infty})$ is sufficient for all Borel sets containing infinity thus defined, but could you please tell me where this definition of extended Borel set is located in this book, or appear in other textbook? | |
| May 7, 2010 at 8:13 | history | answered | Kevin Ventullo | CC BY-SA 2.5 |